The axis of a parabola is along the line y=x and the distance of its vertex A from 0,0 is 2 and that of its focus S from 0,0 is 22. If A and S lie in first quadrant, then the equation of the parabola in parametric form is

TS EAMCET · Maths · Parabola

The axis of a parabola is along the line $y = x$ and the distance of its vertex $A$ from $(0, 0)$ is $\sqrt{2}$ and that of its focus $S$ from $(0, 0)$ is $2 \sqrt{2}$ . If $A$ and $S$ lie in first quadrant, then the equation of the parabola in parametric form is

A $x = {(t + 1)}^{2}, y = {(t - 1)}^{2}$
B $x = t^{2}, y = 2 t$
C $x = {(t - \sqrt{2})}^{2}, y = {(t + \sqrt{2})}^{2}$
D $x = t^{2} + 5, y = t^{2} - 5$

Verified Solution

Answer & Solution

Correct Answer

(A) $x = {(t + 1)}^{2}, y = {(t - 1)}^{2}$

Step-by-step Solution

Detailed explanation

Distance of vertex from origin =2 Distance of focus from origin =22 Axis is along y=x. The focus and the vertex lie in the first quadrant. Hence, the coordinates of the focus are 2,2. The vertex is at 1,1 Hence, the equation of directrix is x+y=0 Take point h,k on the parabola.…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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